Tender and naive weak closure operations on lower BCK-semilattices

The notion of tender and naive weak closure operation is introduced, and their relations and properties are investigated. Using a weak closure operation “cl” and an ideal A of a lower BCK-semilattice X with the greatest element 1, a new ideal K of X containing the ideal A^cl of X is established. Using this ideal K, a new function cl_t: I(X) → I(X), A ↦ K is given, and related properties are considered. We show that if “cl” is a tender (resp., naive) weak closure operation on I(X), then so are “cl_t” and “cl_f”.